Random Structures & Algorithms

On polynomial approximations to AC 0

Early View

Abstract

Classical
AC0 approximation results show that any
AC0 circuit of size
s and depth
d has an
ɛ
‐error probabilistic polynomial over the reals of degree
(log(s/ɛ))O(d). We improve this upper bound to
(logs)O(d)·log(1/ɛ), which is much better for small values of
ɛ. We then use this result to show that
(logs)O(d)·log(1/ɛ)
‐wise independence fools
AC0 circuits of size
s and depth
d up to error at most
ɛ, improving on Tal's strengthening of Braverman's result that
(log(s/ɛ))O(d)
‐wise independence suffices. To our knowledge, this is the first PRG construction for
AC0 that achieves optimal dependence on the error
ɛ. We also prove lower bounds on the best polynomial approximations to
AC0. We show that any polynomial approximating the
OR function on
n bits to a small constant error must have degree at least
Ω∼(logn). This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016).

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